As you can see, 3 different forms are taught for each of the 4 categories above. It explains that there are separate forms for the 1st person singular - Bhithinn, and 1st person plural Bhiomaid (or Bhitheamaid for emphasis), and that Bhiomaid is becoming less common, and particularly in spoken Gàidhlig is being replaced by the regular Bhiodh.

I wanted to clarify if it's definitely ok to replace Bhiomaid with Bhiodh in the spoken language, because I'm looking to sound as natural as possible, and which form is used in writing (both formal and informal writing, that is)? Is Bhiomaid used in writing formal letters and in books etc?

There's a chapter in Blas on short and long forms, I forget which page.

In particular irregular verbs have long (stressed) and short (unstressed) forms. The variations go from hiatus with emphatic endding > hiatus > long vowel > short vowel > reduced vowel > no vowel, though not all verbs have all "stages".
So bhithinn/bhitheamaid are the 2nd most stressed forms, bhiomaid/bhinn are the next step down with /iə/ and /i:/ respectively. Written the same way but one down are /vimɪdʲ/ and /viNʲ/. It's just one of those GOC weirdnesses that they recommend (as far as I recall) the reduced spelling bhiomaid but not bhinn.
Some don't show any spelling variants at all but vary in pronunciation, in tha the vowel goes from long to zero depending on the stress pattern.